On the description of classical reflectionless potentials
The authors consider a one-dimensional system of particles, each having a unit mass with the Hamiltonian \[ H(p,q)=\sum^{n}_{1}p^ 2_ i/2+\sum_{i<j}U(q_ i-q_ j),\quad U'<0,\quad \lim_{x\to 0_+}U(x)=\infty. \] This means that the potential is repulsive and the particles are impenetrable and retain their order on the line. The repulsive potential is called reflectionless if limit velocities of the particles at \(t\to \infty\) can be obtained from the velocities as \(t\to - \infty\) by interchanging the order. Two principal theorems proved here are: (1) Let U be a potential with finite range (i.e. compact support on the right) U(x)\(\equiv 0\) if \(x>x_ 1\). Let U'(x)\(\neq 0\) in the left-half neighborhood of \(x_ 1\). Then U is not a reflectionless classical potential. (2) Let \(a>2\). U be twice continuously differentiable, with U(x)\(\sim c_ 1/x^ a,\quad U'\sim c_ 2/x^{a+1},\quad U''\sim c_ 3/x^{a+2}\) as \(x\to \infty\). Then U(x) is not a reflectionless classical potential.
one-dimensional reflectionless potentials
scientific article; zbMATH DE number 3961028